GCSE Maths Practice: conditional-probability

Question 1 of 10

This question tests conditional probability by excluding one subgroup before calculating the probability.

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\( \begin{array}{l}\text{In a school group, there are 12 Year 10, 8 Year 9, and 10 Year 11 students.} \\ \text{If a student is selected at random, what is the probability they are in Year 10,} \\ \text{given that they are not in Year 11?}\end{array} \)

Choose one option:

Apply the condition first by removing excluded groups before forming the probability.

Conditional Probability Using Group Exclusion

This question focuses on a key Higher GCSE probability skill: calculating probability after applying a condition that removes part of the original group. The phrase given that they are not in Year 11 provides information that must be used before any probability calculation is made.

Conditional probability does not always involve multiple stages or repeated selections. In many cases, such as this one, it involves redefining the sample space to include only outcomes that satisfy a given condition.

Understanding the Sample Space

Before the condition is applied, any student from Year 9, Year 10, or Year 11 could be selected. However, once it is known that the student is not in Year 11, all Year 11 students become impossible outcomes and must be excluded.

The probability must then be calculated using only the remaining students. Using the original total would lead to an incorrect result.

Step-by-Step Method

  1. Identify the total number of individuals in the group.
  2. Apply the condition by removing excluded individuals.
  3. Recalculate the total number of possible outcomes.
  4. Count how many remaining outcomes satisfy the event of interest.
  5. Form the probability as favourable outcomes divided by remaining outcomes.

Worked Example (Different Context)

A sports club has 24 members: 10 juniors, 8 intermediates, and 6 seniors. One member is selected at random. Find the probability that the member is a junior, given that they are not a senior.

The condition removes all senior members from the sample space. The probability must then be calculated using only the remaining members.

Another Example

A survey records participants by transport method: car, bus, or bicycle. If it is known that a selected participant does not travel by bus, probabilities should be calculated using only car and bicycle users.

Common Mistakes

  • Using the original total instead of the reduced total.
  • Failing to remove all excluded outcomes.
  • Interpreting the condition as a second event.
  • Forming the probability before applying the condition.

Why This Is Higher Tier

Although the arithmetic is straightforward, this question tests interpretation and reasoning. Higher-tier GCSE questions often focus on translating written conditions into mathematical logic rather than performing complex calculations.

Real-Life Applications

Conditional probability of this type appears frequently in education data analysis, surveys, and decision-making processes where conclusions are drawn based on restricted groups.

Study Tip

Whenever a probability question includes the phrase given that, rewrite the sample space first. If the condition is applied correctly, the calculation itself becomes simple.

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